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Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.
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%I #12 Oct 15 2024 17:29:25

%S 1,1,1,1,2,1,1,4,3,1,1,8,11,4,1,1,16,43,24,5,1,1,32,171,176,46,6,1,1,

%T 64,683,1376,611,80,7,1,1,128,2731,10944,9281,1864,130,8,1,1,256,

%U 10923,87424,146445,54384,5161,200,9,1,1,512,43691,699136,2334181,1696352,285939,13184,295,10,1

%N Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

%C Two involutions x,y commute if x*y = y*x. Isomorphism is up to permutation of the elements of the (2n)-set. T(n,k) also gives the values for a (2n+1)-set.

%H Andrew Howroyd, <a href="/A362903/b362903.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)

%F G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).

%e Array begins:

%e ======================================================

%e n/k| 0 1 2 3 4 5 6 ...

%e ---+--------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 ...

%e 1 | 1 2 4 8 16 32 64 ...

%e 2 | 1 3 11 43 171 683 2731 ...

%e 3 | 1 4 24 176 1376 10944 87424 ...

%e 4 | 1 5 46 611 9281 146445 2334181 ...

%e 5 | 1 6 80 1864 54384 1696352 53885632 ...

%e 6 | 1 7 130 5161 285939 17562679 1110290303 ...

%e 7 | 1 8 200 13184 1372224 165343616 20774749952 ...

%e 8 | 1 9 295 31532 6101080 1436647664 358238974304 ...

%e ...

%o (PARI) \\ B(n, k) is A022166.

%o B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}

%o C(k,n) = Vec(1/prod(j=0, min(k-1, logint(n, 2)), (1 - x^(2^j) + O(x*x^n))^B(k,j+1), 1 - x + O(x*x^n)))

%o M(n,m=n) = Mat(vector(m+1, k, C(k-1, n)~))

%o { my(A=M(7)); for(i=1, #A, print(A[i,])) }

%Y Columns k=0..3 are A000012, A000027(n-1), A001752, A362904.

%Y Rows n=1..3 are A000079, A007583, A103334(n+1).

%Y Cf. A022166, A362648, A362824, A362826.

%K nonn,tabl

%O 0,5

%A _Andrew Howroyd_, May 11 2023